A276314 G.f. A(x) satisfies: x = A(x)-A(x)^2-3*A(x)^3.

1, 1, 5, 20, 104, 546, 3066, 17655, 104555, 630773, 3867617, 24020932, 150827740, 955808680, 6105327912, 39268000188, 254093573088, 1652984379150, 10804631902350, 70925539707330, 467373389649870, 3090558380977020, 20501504119375500, 136392970090612950

Offset

1,3

Links

Michael De Vlieger, Table of n, a(n) for n = 1..1181

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Thomas M. Richardson, The three 'R's and the Riordan dual, arXiv:1609.01193 [math.CO], 2016.

Formula

G.f.: Series_Reversion(x-x^2-3*x^3)

Conjecture: +169*n*(n+2)*(n-1)*a(n) +13*(n-1) *(13*n^2+26*n-220) *a(n-1) +(-7277*n^3+13423*n^2+43814*n-81700) *a(n-2) -27*(3*n-10) *(3*n-8) *(71*n+197)*a(n-3)=0. - R. J. Mathar, Sep 17 2016

a(n) ~ (29 + 20*sqrt(10))^(n - 1/2) / (2^(5/4) * 5^(1/4) * sqrt(Pi) * n^(3/2) * 13^(n - 1/2)). - Vaclav Kotesovec, Aug 22 2017

Example

G.f.: A(x) = x+x^2+5*x^3+20*x^4+104*x^5+546*x^6+3066*x^7+... Related Expansions:
A(x)^2=x^2+2*x^3+11*x^4+50*x^5+273*x^6+1500*x^7+8664*x^8+...
A(x)^3=x^3+3*x^4+18*x^5+91*x^6+522*x^7+2997*x^8+17831*x^9+...

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - x^2 - 3*x^3, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Aug 22 2017 *)

Prog

(PARI) {a(n)=polcoeff(serreverse(x - x^2 - 3*x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A250886, A276310, A276315, A276316.

Sequence in context: A369488 A092640 A165961 * A292358 A259275 A109500

Adjacent sequences: A276311 A276312 A276313 * A276315 A276316 A276317

Keyword

nonn,easy

Author

Tom Richardson, Aug 29 2016

Status

approved